**Unit 3: Practical Problems in Quantum Mechanics**

*In this unit, you will begin to use the equations and principles of
quantum mechanics to solve some simple problems, involving relatively
simple mathematics. Although the examples presented here involve mostly
“imaginary” particles in ideal quantum systems, these examples will
serve as an introduction to quantum mechanical calculations. The
principles learned using these ideal systems (e.g., particle in a box)
can be used later in “real” systems (molecules). As the system under
study becomes more and more complex, you will learn how to adopt certain
techniques of approximation to simplify your calculation and still
obtain fairly accurate results.*

**Unit 3 Time Advisory**

This unit will take you 16.5 hours to complete.

☐ Subunit 3.1: 6 hours

☐ Subunit 3.2: 6 hours

☐ Subunit 3.3: 4.5 hours

**Unit3 Learning Outcomes**

Upon successful completion of this unit, the student will be able to:
- Explain the difference between classical and quantum mechanics.
- Apply the Schrodinger equation to solve the “particle in a box”
problem.
- Describe quantum phenomena such as tunneling and barrier
penetration.
- Derive energy levels for quantized systems, such as a harmonic
oscillator.
- Describe the molecular vibrations in terms of simple systems, such
as a spring.
- Describe the difference between harmonic and anharmonic oscillators.
- Transform equation from Cartesian coordinates to polar and/or
spherical coordinates.
- Solve the Schrodinger equation in polar and spherical coordinates.
- Explain how the angular momentum is a conserved quantity.
- Describe the motion of a particle around a ring or on the surface of
a sphere.
- Apply angular momentum operators to solve “spin” problems.

**3.1 Particle in a Box (Translational Motion)**
- **Reading: Boston University: Professor Dan Dill’s “Analytic
Solution of the Schrödinger Equation: Particle in a Box” and
“Example of One-Dimensional Quantum System”**
Link: Boston University: Professor Dan Dill’s “Analytic Solution of
the Schrödinger Equation: Particle in a
Box” and
“Example of One-Dimensional Quantum
System” (PDF)

Instructions: Please click on the first and second links above and
scroll down the webpage to the italic headings, “*Particle in a
Box*” and “*Example of One-Dimensional Quantum System*.” To open
each set of notes, click on the hyperlink next to these italicized
headings. A PDF file of each section will open up. Read each PDF
file in its entirety (7 pages and 6 pages, respectively). While
reading the material, repeat each mathematical derivation on scratch
paper. Studying this resource should take approximately 2 hours to
complete. Note that this reading also covers the material you need
to know for subunits 3.1.1–3.1.4.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

**Reading: The Chemistry Hypermedia Project’s “Particle in a Box”**Link: The Chemistry Hypermedia Project’s “Particle in a Box” (HTML)

Instructions: Please click on the link and read the Chemistry Hypermedia Project webpage, which provides a glance of the particle in a box problem. While reading the material, repeat each mathematical derivation on scratch paper. Studying this resource should take approximately 1.5 hours to complete. Note that this reading also covers the material you need to know for subunits 3.1.1–3.1.4.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Reading: Everyscience.com’s “Particle in a Two-Dimensional Box”**Link: Everyscience.com’s “Particle in a Two-Dimensional Box” (HTML)

Instructions: Please click on the link and read the entire webpage, which presents the problem in 2D. While reading the material, repeat each mathematical derivation on scratch paper. Studying this resource should take approximately 1 hour to complete. Note that this reading also covers the material you need to know for subunits 3.1.1–3.1.4.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Web Media: Kansas State University: Physics Education Research Group’s “Visual Quantum Mechanics – Quantum Tunneling”**Link: Kansas State University: Physics Education Research Group’s “Visual Quantum Mechanics – Quantum Tunneling” (Adobe Shockwave)

Instructions: Please click on the link above and follow the steps on the webpage to simulate*tunneling*. Studying this resource should take approximately 1 hour to complete. Please note that this program requires Adobe Shockwave. If you do not already have Shockwave, you can download a free version here. Note that this reading also covers the material you need to know for subunits 3.1.1–3.1.4.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Web Media: YouTube: Praba Siva’s “Schrodinger Equation – For Particle in a 1 D, 2D, 3D Box” and David Colarusso’s “What Is Quantum Tunneling?”**Links: YouTube: Praba Siva’s “Schrodinger Equation – For Particle in a 1 D, 2D, 3D Box” and David Colarusso’s “What Is Quantum Tunneling?” (YouTube)

Instructions: Please click on the links above, and watch these videos. These are short videos that explain how to solve the Schrodinger equation for one-, two-, and three-dimensional systems (Praba Siva’s video) and explain the phenomenon of quantum tunneling (David Colarusso’s video). Note that these videos also cover the material you need to know for subunits 3.1.1–3.1.4. Studying these resources should take approximately 0.5 hours to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Assessment: The Saylor Foundation’s “Assessment 5”**Link: The Saylor Foundation’s “Assessment 5” (DOC)

Instructions: Complete the attached assessment questions to check your understanding of the material covered thus far. Once you have completed the assessment, you may check your answers against the “Answer Key” (DOC).

Completing this assessment should take approximately 1 hour.

**3.1.1 Particle in a One-Dimensional Box**
*Note: This topic is covered by the readings assigned beneath subunit
3.1. In particular, please focus on Chemistry Hypermedia Project’s
“Particle in a Box” and Professor Dan Dill’s “Analytic Solution of the
Schrödinger Equation: Particle in a Box” to learn about one-dimensional
quantum mechanical systems.*

**3.1.2 Example of One-Dimensional Quantum Systems**
*Note: This topic is covered by the readings assigned beneath subunit
3.1. In particular, please focus on Professor Dan Dill’s “Example of
One-Dimensional Quantum System” to learn about the behavior of quantum
particles “confined” into quantum wells with different energy barriers.*

**3.1.3 Barrier Penetration and Tunneling**
*Note: This topic is covered by the readings assigned beneath subunit
3.1. In particular, please focus on David Colarusso’s “What Is Quantum
Tunneling?” and Professor Dan Dill’s “Example of One-Dimensional Quantum
System” to learn about the unique phenomenon of quantum tunneling.*

**3.1.4 Particle in a 2D or 3D Box**
*Note: This topic is covered by the readings assigned beneath subunit
3.1. In particular, please focus on Everyscience.com’s “Particle in a
Two-Dimensional Box” and Praba Siva’s web media, “Schrodinger
Equation – For Particle in a 1 D, 2D, 3D Box,” to learn about the
behavior of quantum particles in more than one dimension.*

**3.2 Vibrational Motion**
- **Reading: Everyscience.com’s “Molecular Vibrations” and “Anharmonic
Oscillator”**
Links: Everyscience.com’s “Molecular
Vibrations”
and “Anharmonic
Oscillator”
(HTML)

Instructions: Please click on the “Molecular Vibrations” and
“Anharmonic Oscillator” links and read these webpages in their
entirety. While reading the material, repeat the mathematical
derivation of the vibrational terms. Studying this resource should
take approximately 1.5 hours to complete. Note that these readings
also cover the materials you need to know for subunits
3.2.1–3.2.3.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

**Reading: Boston University: Professor Dan Dill’s “Harmonic Oscillator”**Link: Boston University: Professor Dan Dill’s “Harmonic Oscillator” (PDF)

Instructions: For Professor Dill’s notes, please click on the link above, scroll down the webpage to the italic heading, “*Harmonic Oscillator*,” and click on the hyperlink to the PDF next to the heading. Read the entire PDF (18 pages). While reading the material, repeat the mathematical derivation of the harmonic potential and understand the difference between harmonic and anharmonic vibrational terms. Studying this resource should take approximately 3 hours to complete. Note that this reading also covers the materials you need to know for subunits 3.2.1–3.2.3.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Web Media: YouTube: Indian Institute of Technology, Madras: Professor K. Mangala Sunder’s “Lecture 4 Harmonic Oscillator and Molecular Vibration”**Link: YouTube: Indian Institute of Technology, Madras: Professor K. Mangala Sunder’s “Lecture 4 Harmonic Oscillator and Molecular Vibration” (YouTube)

Instructions: Please click on the link above, and watch the entire video. While watching the video, repeat the mathematical derivation following the professor’s notes on the board. Studying this resource should take approximately 1.5 hours to complete. Note that this video lecture also covers the material you need to know for subunits 3.2.1–3.2.3.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Assessment: The Saylor Foundation’s “Assessment 6”**Link: The Saylor Foundation’s “Assessment 6” (DOC)

Instructions: Complete the attached assessment questions to check your understanding of the material covered thus far. Once you have completed the assessment, you may check your answers against the “Answer Key” (DOC).

Completing this assessment should take approximately 1 hour.

**3.2.1 Overview of Molecular Vibrations**
*Note: This subunit is covered by the readings assigned beneath subunit
3.2. In particular, please focus on Everyscience.com’s “Molecular
Vibrations” to have an overview of molecular vibrations.*

**3.2.2 Harmonic Oscillator**
*Note: This subunit is covered by the readings assigned beneath subunit
3.2. In particular, please focus on Professor Dan Dill’s “Harmonic
Oscillator” and Professor K. Mangala Sunder’s “Lecture 4 Harmonic
Oscillator and Molecular Vibration” to learn how harmonic oscillators
can be used to model molecular vibrations.*

**3.2.3 The Anharmonic Oscillator**
*Note: This subunit is covered by the readings assigned beneath subunit
3.2. In particular, please focus on Everyscience.com’s “Anharmonic
Oscillator” to learn how the anharmonic oscillator is a better
representation of molecular vibration as it allows bond dissociation at
high vibrational excitations.*

**3.3 Angular Momentum and Rotational Motion**
- **Web Media: bpReid-Software for Science and Mathematics’ “Particle
on a Sphere – Spherical Harmonics”**
Link: bpReid-Software for Science and Mathematics’ “Particle on a
Sphere – Spherical Harmonics”
(HTML)

Instructions: Please click on the link above, and complete the
exercises on the webpage to simulate the *spherical harmonics*.
Please note that this program requires Oracle’s Java Runtime. If
you do not already have it, you can download a free version
here. Studying this
resource should take approximately 0.5 hours to complete. Note that
this resource also covers the material you need to know for subunits
3.3.1–3.3.5.

Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.

**Reading: Everyscience’s “Polar Coordinates”**Link: Everyscience’s “Polar Coordinates” (HTML)Instructions: Please click on link to learn about polar coordinates. In this resource, it is important that you learn how to transform Cartesian coordinates into polar coordinates. Studying this resource should take approximately 0.5 hours to complete. Note that this reading also covers the material you need to know for subunits 3.3.1–3.3.5.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Reading: Boston University: Professor Dan Dill’s “A Little Bit of Angular Momentum,” “Angular Motion in Two-Components Systems,” and “Particle Moving on a Ring”**Link: Boston University: Professor Dan Dill’s “A Little Bit of Angular Momentum”, “Angular Motion in Two-Components Systems”, and “Motion of a Particle on a Ring” (PDF)

Instructions: For Professor Dill’s notes, please click on the first and second link above and scroll down to the italic headings, “*A little Bit of Angular Momentum*” and “*Angular Motion in Two-Components Systems*.” To download the PDF file, click on the hyperlinks next to the heading. Read the entire PDF files (4 pages and 18 pages, respectively). In this resource, you will use polar coordinates to solve the Schrodinger equation and derive parameters related to angular motion. Next, click on the “*Motion of a Particle on a Ring*” link and read the entire webpage, which offers an animated representation of the probability density of a particle moving on a ring. Studying these resources should take approximately 2 hours to complete. Note that these readings also cover the material you need to know for subunits 3.3.1–3.3.5.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Reading: University of California Davis: UC Davis ChemWiki’s “Particle on a Ring” and “Particle on a Sphere”**Links: University of California Davis: UC Davis ChemWiki’s “Particle in a Ring” and “Particle in a Sphere” (HTML)

Instructions: Please click on the UC Davis ChemWiki links for a quick overview of a particle moving in a ring and in a sphere. Studying these resources should take approximately 0.5 hours to complete. Note that these readings also cover the material you need to know for subunits 3.3.1–3.3.5.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Web Media: YouTube: Yale Courses: “Quantum Mechanics III”**Link: YouTube: Yale Courses: “Quantum Mechanics III” (YouTube)Also Available in:

HTML transcript, Mp3 audio, Mp4 video, Adobe Flash video from Open Yale Courses

Instructions: Please click on the link above, and watch the video. You can focus on the lecture portion starting from approximately minute 12:40 to learn about the quantum motion of a particle on a ring. Studying this resource should take approximately 1 hour to complete.

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.**Assessment: The Saylor Foundation’s “Assessment 7”**Link: The Saylor Foundation’s “Assessment 7” (DOC)

Instructions: Complete the attached assessment questions to check your understanding of the material covered thus far. Once you have completed the assessment, you may check your answers against the “Answer Key” (DOC).

Completing this assessment should take approximately 1 hour.

**3.3.1 Angular Momentum**
*Note: This subunit is covered by the readings assigned beneath subunit
3.3. In particular, please focus on Professor Dan Dill’s “Angular
Motion in Two-Components Systems” to learn about angular momentum.*

**3.3.2 Polar and Spherical Coordinates**
*Note: This subunit is covered by the readings assigned beneath subunit
3.3. In particular, please focus on and Everyscience’s “Polar
Coordinates” to learn how to transform Cartesian coordinates into polar
and spherical coordinates.*

**3.3.3 Particle on a Ring**
*Note: This subunit is covered by the readings assigned beneath subunit
3.3. In particular, please focus on Professor Dan Dill’s “Particle
Moving on a Ring,” ChemWiki’s “Particle on a Ring,” and the web media
Yale courses, “Quantum Mechanics III,” from approximately minute 12:40,
to learn about the quantum motion of a particle on a ring.*

**3.3.4 Particle on a Sphere**
*Note: This subunit is covered by the readings assigned beneath subunit
3.3. In particular, please focus on UC Davis ChemWiki’s “Particle on a
Ring and Particle on a Sphere” and on the Web Media: “Particle on a
Sphere – Spherical Harmonics.”*

**3.3.5 Spin**
*Note: This subunit is covered by the readings assigned beneath subunit
3.3. In particular, please focus on Professor Dan Dill’s “A Little Bit
of Angular Momentum” to learn about spin angular momentum.*